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Theorem cbv1 2431
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) Format hypotheses to common style. (Revised by Wolf Lammen, 13-May-2018.)
Hypotheses
Ref Expression
cbv1.1 𝑥𝜑
cbv1.2 𝑦𝜑
cbv1.3 (𝜑 → Ⅎ𝑦𝜓)
cbv1.4 (𝜑 → Ⅎ𝑥𝜒)
cbv1.5 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
Assertion
Ref Expression
cbv1 (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))

Proof of Theorem cbv1
StepHypRef Expression
1 cbv1.2 . . . . 5 𝑦𝜑
2 cbv1.3 . . . . 5 (𝜑 → Ⅎ𝑦𝜓)
31, 2nfim1 2224 . . . 4 𝑦(𝜑𝜓)
4 cbv1.1 . . . . 5 𝑥𝜑
5 cbv1.4 . . . . 5 (𝜑 → Ⅎ𝑥𝜒)
64, 5nfim1 2224 . . . 4 𝑥(𝜑𝜒)
7 cbv1.5 . . . . . 6 (𝜑 → (𝑥 = 𝑦 → (𝜓𝜒)))
87com12 32 . . . . 5 (𝑥 = 𝑦 → (𝜑 → (𝜓𝜒)))
98a2d 29 . . . 4 (𝑥 = 𝑦 → ((𝜑𝜓) → (𝜑𝜒)))
103, 6, 9cbv3 2429 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑦(𝜑𝜒))
11419.21 2234 . . 3 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
12119.21 2234 . . 3 (∀𝑦(𝜑𝜒) ↔ (𝜑 → ∀𝑦𝜒))
1310, 11, 123imtr3i 281 . 2 ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑦𝜒))
1413pm2.86i 110 1 (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1632  wnf 1859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-ex 1856  df-nf 1861
This theorem is referenced by:  cbv1h  2432
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